Incline Plane Problem: Determining Velocity of a Mass at Height h1

[Benzoate]

Homework Statement

A mass slides on a frictionless plane inclined at an angle theta from the horizontal. The mass starts from rest at a height h(1)+h(2) , then slides off the ramp at height h(1).

I will post the link so you can visualized the problem

http://courses.ncsu.edu/py411/lec/001/

What is the velocity vector of the mass when the mass is at height h1 (as the mass leaves the ramp?)

Homework Equations

Force equations , Mechanical Energy equations

The Attempt at a Solution

I apply to methods to determined the velocity of mass : Forces equations and Mechanical Energy equations. I will start with Force equations

y component: F(normal)-mg cos(theta)=0, nothing moves in y direction

x-components: mg sin(theta)-0=m*a=> a=g*sin(theta) ; next thing I did was integrate a to get v and now v = gt*sin(theta). My prof said he wanted my v in vector form, meaning I think he wants the velocity , in the x, y and z directions. so v= (gt sin(theta))x + (gt cos(theta)+0 z.

Now I will find the velocity the alternative way:

(K(f)-K(i))+(U(f)-U(i))=0=> ((.5*m*v^2)-0)+(0-m*g*h2))=0 +> v=sqrt(2*g*h2). Both methods lead me to different velocities. What did I do wrong?

 

[Doc Al]

x-components: mg sin(theta)-0=m*a=> a=g*sin(theta) ; next thing I did was integrate a to get v and now v = gt*sin(theta).

You found the speed as a function of time (assuming it starts from rest). What you need is the speed at the end of the ramp. Keep going.

And note that that velocity will be parallel to the ramp, so find its components properly.

 

[Benzoate]

You found the speed as a function of time (assuming it starts from rest). What you need is the speed at the end of the ramp. Keep going.

And note that that velocity will be parallel to the ramp, so find its components properly.

should I continue to integrate the velocity vector to come up with the position vector? Now I have an equation that looks like this:

gt^2/2*sin(theta)=x, x being the length of the ramp I guess

gt^2/2*(h2/sqrt((h2)^2+x^2))=sqrt((h2)^2+x^2)) => t= sqrt(2*((h2)^2+x^2)/g*h2)) and now I can plug t into v=gt*sin(theta) right?

 

[Doc Al]

Right idea. But you want to eliminate x, so rewrite it in terms of h2 and theta. Then be sure to simplify the final expression, so you can compare it with your other solution.